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Summary

DescriptionBifurcation1-2.png	English: Bifurcation of periodic points from period 1 to 2 for fc(z)=z*z +c. Parabolic parameter c = -3/4 and fixed point z = 1/2
Date	20 June 2009
Source	Own work
Author	Adam majewski
Other versions	image from Bifurcations of One-Dimensional One-Parametric Maps Revisited by Lenka Přibylová image from Example 2 from math.stackexchange question: does-complex-dynamics-offer-any-insights-into-real-dynamics ?

Summary

This image shows some features of the discrete dynamical system

z_{n+1}=f_{c}(z_{n})\,

based on complex quadratic polynomial :

f_{c}(z)=z^{2}+c.\,

.

When coefficient $c\,$ goes from c=0.25 to c=-2 along horizontal axis ( imaginary part of c is zero and it is a 3D diagram of function which gets real input and gives complex output) then limit cycle is changing from fixed point ( period 1) to period 2 cycle. This qualitative change is called bifurcation.

This path is inside Mandelbrot set ( escape route). It s also first of period doubling bifurcation.

Note that :

there are fixed points for all c values, but they change from attracting to indifferent( in parabolic point, root point) and repelling
there are 2 period 2 points for all c values. They also change from from attracting to indifferent( in parabolic point, root point) and repelling.
in bifurcation point ( root, parabolic) all period 2 values and fixed point have the same value and the same (=1) stability index .
before and after bifurcation point period 2 points creates 3D parabolas, which are rotated ( 90 degrees) with respect to themselves

stability index of period 1 points	period 1 points on dynamic plane	period 1 points on parameter plane
changes from attractive through indifferent to repelling	moves from interior of Kc to its boundary	moves from interior of component of M-set to its boundary

Please check demo 2 page 3 from program Mandel by Wolf Jung to see another visualisation of this bifurcation.

dynamics

evolution of dynamics along escape route 0 ( parabolic implosion)
parameter c	location of c	Julia set	interior	type of critical orbit dynamics	critical point	fixed points	stability of alfa
c = -3/4	boundary, root point	connected	exist	parabolic	attracted to alfa fixed point	alfa fixed point equal to beta fixed point, both are parabolic	r = 1
0 < x < -3/4	internla ray 1/2	connected	exist	attracting	attracted to alfa fixed point		0 < r < 1.0
c = 0	center, interior	connected = Circle Julia set	exist	superattracting	attracted to alfa fixed point	fixed critical point equal to alfa fixed point, alfa is superattracting, beta is repelling	r = 0
0<c<1/4	internal ray 0, interior	connected	exist	attracting	attracted to alfa fixed point	alfa is attracting, beta is repelling	0 < r < 1.0
c = 1/4	cusp, boundary	connected = cauliflower	exist	parabolic	equal to alfa fixed point	alfa fixed point equal to beta fixed point, both are parabolic	r = 1
c>1/4	external ray 0, exterior	disconnected = imploded cauliflower	disappears	repelling	repelling to infinity	both finite fixed points are repelling	r > 1

Stability r is absolute value of multiplier m at fixed point alfa $z_{\alpha }$ :

 $r=|m(z_{\alpha })|$

c = 0.0000000000000000+0.0000000000000000*I 	 m(c) = 0.0000000000000000+0.0000000000000000*I 	 r(m) = 0.0000000000000000 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.0250000000000000+0.0000000000000000*I 	 m(c) = 0.0513167019494862+0.0000000000000000*I 	 r(m) = 0.0513167019494862 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.0500000000000000+0.0000000000000000*I 	 m(c) = 0.1055728090000841+0.0000000000000000*I 	 r(m) = 0.1055728090000841 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.0750000000000000+0.0000000000000000*I 	 m(c) = 0.1633399734659244+0.0000000000000000*I 	 r(m) = 0.1633399734659244 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.1000000000000000+0.0000000000000000*I 	 m(c) = 0.2254033307585166+0.0000000000000000*I 	 r(m) = 0.2254033307585166 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.1250000000000000+0.0000000000000000*I 	 m(c) = 0.2928932188134524+0.0000000000000000*I 	 r(m) = 0.2928932188134524 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.1500000000000000+0.0000000000000000*I 	 m(c) = 0.3675444679663241+0.0000000000000000*I 	 r(m) = 0.3675444679663241 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.1750000000000000+0.0000000000000000*I 	 m(c) = 0.4522774424948338+0.0000000000000000*I 	 r(m) = 0.4522774424948338 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.2000000000000000+0.0000000000000000*I 	 m(c) = 0.5527864045000419+0.0000000000000000*I 	 r(m) = 0.5527864045000419 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.2250000000000000+0.0000000000000000*I 	 m(c) = 0.6837722339831620+0.0000000000000000*I 	 r(m) = 0.6837722339831620 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.2500000000000000+0.0000000000000000*I 	 m(c) = 0.9999999894632878+0.0000000000000000*I 	 r(m) = 0.9999999894632878 	 t(m) = 0.0000000000000000 	period = 1
 c = 0.2750000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.3162277660168377*I 	 r(m) = 1.0488088481701514 	 t(m) = 0.0487455572605341 	period = 1
 c = 0.3000000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.4472135954999579*I 	 r(m) = 1.0954451150103321 	 t(m) = 0.0669301182003075 	period = 1
 c = 0.3250000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.5477225575051662*I 	 r(m) = 1.1401754250991381 	 t(m) = 0.0797514300099943 	period = 1
 c = 0.3500000000000000+0.0000000000000000*I 	 m(c) = 1.0000000000000000+0.6324555320336760*I 	 r(m) = 1.1832159566199232 	 t(m) = 0.0897542589928440 	period = 1

==Maxima CAS src code==
<pre>
 GiveRoots_bf(g):=
 block(
  [cc:bfallroots(expand(g)=0)],
  cc:map(rhs,cc),/* remove string "c=" */
  return(cc)
 )$ 


 /* functions for computing periodic points ;   */ 
 give_beta(_c):= (1+sqrt(abs(1-4*_c)))/2 $  
 give_alfa(_c):= (1-sqrt(abs(1-4*_c)))/2 $ 
 give_2(c):=
 block(
 [eq,rr],
 eq:z*z +z +c +1,  
 rr:GiveRoots_bf(eq), 
 return(float(rr))
 ); 


 xMax:0;
 xMin:-1.39;
 yMin:-2;
 yMax:2;
 iXmax:1000;
 dx:(xMax-xMin)/iXmax;

 /*  points */
 p_pts:[ [-0.75,-0.5,0] ];
 p1_beta:[];
 p1_alfa_r:[];
 p1_alfa_a:[];
 p2_r:[]; /* period 2 repelling */
 p2_a:[]; /* period 2 attracting */ 


 /* -------------------- main  ----------------------- */
 for c:xMin step dx thru xMax do 
 (
 alfa:give_alfa(c),
 if cabs(2*alfa)>1 
  then p1_alfa_r:cons([c,realpart(alfa),imagpart(alfa)],p1_alfa_r)
  else p1_alfa_a:cons([c,realpart(alfa),imagpart(beta)],p1_alfa_a),
 roots:allroots(z*z +z +c +1=0),
 z2:rhs(roots[1]),
 if cabs(float(4*z2*(z2*z2+c)))>1  /* multiplier */
  then for z in roots do p2_r:cons([c,realpart(rhs(z)),imagpart(rhs(z))],p2_r)
  else for z in roots do p2_a:cons([c,realpart(rhs(z)),imagpart(rhs(z))],p2_a)
 );


 load(cpoly); /* for bfallroots */
 load(draw);


 draw3d(
  terminal = screen,
  pic_height= iXmax,
  title       = "periodic z-points for c along horizontal axis  for fc(z)= z*z +c ",
  ylabel     = "Re(z)",
  zlabel ="Im(z)",
  xlabel     = "c-coefficient",
  yrange = [yMin,yMax],
  point_type    = filled_circle,
  point_size    = 0.2,
  points_joined = true,
  /* period 1 */
  key    = " alfa repelling",
  color  = dark-blue,
  points(p1_alfa_r),
  key    = " alfa attracting",
  color  = light-blue,
  points(p1_alfa_a),
  /* period 2 */
  points_joined = false,
  key    = " period 2 attracting",
  color  = dark-green,
  points(p2_a),
  key    = " period 2 repelling",
  color  = light-green,
  points(p2_r),
  /* grid and tics */
  xtics      = {-3/4}, 
  /* -2,root points,centers, 0 */
  /*xtics_axis = true,  plot tics on x-axis */
  xtics_rotate = true,
  ytics      = {-0.5},
  ztics = {-1,0,1},
  grid       = true, /* draw grid*/
  /* special points */
  point_size    = 0.7,
  color         = red,
  key           = "bifurcation",
  points(p_pts)
 )$

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Summary

Summary

dynamics

Licensing